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Difficulty: HardExponents, Roots, and Scientific Notation

A supercomputer simulation requires a total of 1.62×10141.62 \times 10^{14} operations. The simulation is divided equally among 12 identical processors running in parallel. If each processor can perform 4.5×1094.5 \times 10^9 operations per second, how many seconds will it take to complete the simulation?

  1. A
    6.075×10226.075 \times 10^{22}
  2. B
    3.6×1043.6 \times 10^4
  3. 3.0×1033.0 \times 10^3Answer
  4. D
    4.32×1054.32 \times 10^5
  5. E
    3.0×1043.0 \times 10^4

Answer

3.0×1033.0 \times 10^3
The correct answer is 3.0×1033.0 \times 10^3. To find the total time to complete the simulation, first compute the total operations the 12 parallel processors can perform per second: 12×(4.5×109)=5.4×101012 \times (4.5 \times 10^9) = 5.4 \times 10^{10} operations/second. Next, divide the total number of required operations by this combined rate: 1.62×10145.4×1010=0.3×1014−10=0.3×104=3.0×103\frac{1.62 \times 10^{14}}{5.4 \times 10^{10}} = 0.3 \times 10^{14-10} = 0.3 \times 10^4 = 3.0 \times 10^3 seconds.

Step-by-Step Solution

1
Determine the combined processing rate of the 12 processors working in parallel.
Combined Rate = 12×(4.5×109)=5.4×101012 \times (4.5 \times 10^9) = 5.4 \times 10^{10} operations per second.
Since the 12 processors run in parallel, their individual speeds are added together to find the total rate of operations performed per second.
2
Divide the total number of operations required by the combined rate of all processors to find the time in seconds.
Time = 1.62×10145.4×1010\frac{1.62 \times 10^{14}}{5.4 \times 10^{10}}
Time equals the total workload divided by the combined rate of work.
3
Perform the division and convert the result to standard scientific notation.
Time = 0.3×1014−10=0.3×104=3.0×1030.3 \times 10^{14 - 10} = 0.3 \times 10^4 = 3.0 \times 10^3 seconds.
Subtracting the exponents during division gives 10410^4. The coefficient 0.30.3 is rewritten in standard scientific notation as 3.0×10−13.0 \times 10^{-1}, so 0.3×104=3.0×1030.3 \times 10^4 = 3.0 \times 10^3.

Key Concept

Scientific notation operations and rate division
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Topics

Exponents, Roots, and Scientific Notation

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